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Quasi-decidability of a Fragment of the First-Order Theory of Real Numbers

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Abstract

In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of n equations in n variables, and for which all functions are computable in the sense that it is possible to compute arbitrarily close interval approximations. Even though this fragment is undecidable, we prove that—under the additional assumption of bounded domains—there is a (possibly non-terminating) algorithm for checking satisfiability such that (1) whenever it terminates, it computes a correct answer, and (2) it always terminates when the input is robust. A formula is robust, if its satisfiability does not change under small continuous perturbations. We also prove that it is not possible to generalize this result to the full first-order language—removing the restriction on the number of equations versus number of variables. As a basic tool for our algorithm we use the notion of degree from the field of topology.

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Notes

  1. This follows from the fact that X resp. Y can be separated by open \(\varepsilon '\)-neighborhoods U(X) resp. U(Y) with positive distance from each other, and the fact that using the uniform continuity of |f| and g, \(X'\subseteq U(X)\) and \(Y'\subseteq U(Y)\) for \(\alpha \) small enough.

  2. The set \(\{(p,x) \mid g(p,x)>\alpha /4\}\) is an open neighborhood of \(\{p_0\}\times \bar{{\varOmega }}\) and the compactness of \(\bar{{\varOmega }}\) implies that there is a neighborhood \(U(p_0)\) of \(\{p_0\}\) such that \(U(p_0)\times {\varOmega }\subseteq \{(p,x) \mid g(p,x)>\alpha /4\}\).

  3. That is, it may call \(\mathcal {I}(f)\) with any input an arbitrary number of times, but apart from the results of calling \(\mathcal {I}(f)\) it does not use any properties of f, nor does it analyze how \(\mathcal {I}(f)\) is computed.

  4. This can be shown as follows. The function \(\tilde{f}:= f/|f|{:} \partial B\rightarrow S^{n-1}\) is homotopic to \(g(x):=\frac{\tilde{f}(x)-\tilde{f}(-x)}{|\tilde{f}(x)-\tilde{f}(-x)|}\) via the homotopy \(H(t,x)=\frac{\tilde{f}(x)-t\tilde{f}(-x)}{|\tilde{f}(x)-t\tilde{f}(-x)|}\), so \(\tilde{f}\) and g have the same degree. Assumptions on B imply that \(\partial B\simeq S^{n-1}\) and an odd map \(g(-x)=-g(x)\) between spheres has odd degree [14, p. 180].

  5. http://topdeg.sourceforge.net.

  6. http://chomp.rutgers.edu.

  7. http://www.linalg.org/gap.html.

  8. http://comptop.stanford.edu/u/programs/plex/.

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Acknowledgments

The work of Stefan Ratschan and Peter Franek was supported by MŠMT Project Number OC10048 and the Czech Science Foundation (GACR) Grants Number P202/12/J060 and 15-14484S with Institutional Support RVO:67985807.

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Authors and Affiliations

  1. Institute of Computer Science, The Czech Academy of Sciences, Prague, Czech Republic

    Peter Franek & Stefan Ratschan

  2. Jagellonian University, Kraków, Poland

    Piotr Zgliczynski

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  1. Peter Franek
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  2. Stefan Ratschan
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  3. Piotr Zgliczynski
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Correspondence to Stefan Ratschan.

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This is an extended and revised version of a paper that appeared in the proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science [18]

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Franek, P., Ratschan, S. & Zgliczynski, P. Quasi-decidability of a Fragment of the First-Order Theory of Real Numbers. J Autom Reasoning 57, 157–185 (2016). https://doi.org/10.1007/s10817-015-9351-3

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